James Maddison scored twice, his first after 36 seconds and his second on the stroke of halftime, while Son Heung-min, Dejan Kulusevski and Pape Matar Sarr were also on target in the away game on Sunday.
As Southampton imploded, many of their fans left well before the interval, although those that stayed at least watched their team recover a modicum of pride in the second half.
Victory was a welcome boost for Spurs, who had managed only one point from their last three league games. They moved up to 10th with 23 points from 16 games.
Southampton remain on five points and already look destined for a rapid return to the Championship as they languish nine points from the safety zone.
Injury-hit Tottenham have looked fragile in recent weeks, winning only once in eight games in all competitions, but from the moment Djed Spence's sublime pass sent Maddison clear to fire the opener the confidence flooded back.
They were helped by a shocking first-half display by Russell Martin's Southampton side.
Son swept in the second after 12 minutes and a couple of minutes later the South Korean's pass into the area fell for Kulusevski to score from close range.
Sarr showed great footwork to slot home Tottenham's fourth goal by which time the stands were rapidly emptying.
Maddison netted his second in first-half stoppage time, latching onto Son's pass and scoring from a tight angle.
Spurs took their foot off the gas after the break and ended the game with four teenagers on the pitch as manager Ange Postecoglou rung the changes ahead of a League Cup quarter-final at home to Manchester United on Thursday.
